2020/12/01 by Marco Peruzzetto, Peruzzetto, Marco
Mathematics · Computer Science · #Advanced Banach Space Theory #Optimization and Variational Analysis #Nonlinear Differential Equations Analysis
paper · pdf · doi:10.48550/arxiv.2012.00615
Let L(X;Y) be the space of bounded linear operators from a Banach space X to a Banach space Y. Given an operator-valued function u:ℝ≥ 0→ L(X;Y), suppose that every orbit t↦ u(t)x has a regularity property (e.g. continuity, differentiability, etc.) on some interval (tx,∞) in general depending on x∈ X. In this paper we develop an abstract set-up based on Baire-type arguments which allows, under certain conditions, removing the dependency on x systematically. Afterwards, we apply this theoretical framework to several different regularity properties that are of interest also in semigroup theory. In particular, a generalisation of the prior results on eventual differentiability of strongly continuous functions u:ℝ≥ 0 → L(X;Y) obtained by Iley and Bárta follows as a special case of our method.